The quadratic form index 
associated to a symmetric,
nondegenerate bilinear
form 
on a finite-dimensional real vector space 
is a nonnegative
integer defined by
 |
where the set 
is defined to be
 |
As a concrete example, a pair 
consisting of a smooth
manifold 
with a smooth, nondegenerate, symmetric
real 
tensor field 
is said to be a Lorentzian
manifold iff 
and the index 
associated to the bilinear form 
satisfies 
for all 
(Sachs and Wu 1977). This particular definition succinctly
conveys the fact that Lorentzian manifolds have indefinite metric
tensors of signature 
in the convention counting negative directions first
without having to make precise any definitions related to metric signatures, quadratic
form signatures, etc.
The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index
of a metric tensor 
defined on a smooth manifold 
. In particular, the index of a metric
tensor 
at a point 
is defined to be the quadratic form index associated
to the bilinear form 
. Because of this connection, indices are especially significant
to various fields: In some literature on differential
geometry and Riemannian geometry, for
example, the notion of an index is used as a main tool by which to define
a metric tensor (Sachs and Wu 1977).
